A Practical Introduction to Data Structures and Algorithm Analysis

186 Chap. 5 Binary Trees172 34645 6 7123 5(a)1723564 5 6 74 2 1 3(b)Figure 5.20 Two series of exchanges to build a max-heap. (a) This heap is builtby a series of nine exchanges in the order (4-2), (4-1), (2-1), (5-2), (5-4), (6-3),(6-5), (7-5), (7-6). (b) This heap is built by a series of four exchanges in the order(5-2), (7-3), (7-1), (6-1).RH 1 H 2Figure 5.21 Final stage in the heap-building algorithm. Both subtrees of node Rare heaps. All that remains is to push R down to its proper level in the heap.array, with the first internal node. The exchanges shown in Figure 5.20(b) resultfrom this process. Method buildHeap implements the building algorithm.What is the cost of buildHeap? Clearly it is the sum of the costs for the callsto siftdown. Each siftdown operation can cost at most the number of levels ittakes for the node being sifted to reach the bottom of the tree. In any complete tree,approximately half of the nodes are leaves and so cannot be moved downward atall. One quarter of the nodes are one level above the leaves, and so their elementscan move down at most one level. At each step up the tree we get half the number ofnodes as were at the previous level, and an additional height of one. The maximum